Two dimensional QCD is a one dimensional Kazakov-Migdal model

TL;DR

This paper derives a one-dimensional matrix model representation for 2D QCD partition functions on a cylinder and torus, revealing a fermionic interpretation and connecting gauge theory to integrable models.

Contribution

It explicitly computes 2D QCD partition functions and shows their equivalence to a Kazakov-Migdal matrix model with eigenvalues on a circle, including a non-trivial modular transformation.

Findings

Partition functions match standard representation expansions

States interpreted as a gas of free fermions

Explicit grand canonical partition function derived

Abstract

We calculate the partition functions of QCD in two dimensions on a cylinder and on a torus in the gauge $\partial_{0} A_{0} = 0$ by integrating explicitly over the non zero modes of the Fourier expansion in the periodic time variable. The result is a one dimensional Kazakov-Migdal matrix model with eigenvalues on a circle rather than on a line. We prove that our result coincides with the standard expansion in representations of the gauge group. This involves a non trivial modular transformation from an expansion in exponentials of $g^2$ to one in exponentials of $1/g^2$. Finally we argue that the states of the $U(N)$ or $SU(N)$ partition function can be interpreted as a gas of N free fermions, and the grand canonical partition function of such ensemble is given explicitly as an infinite product.